\(3^{15}-9^6=3^{15}-3^{12}=3^{12}\left(3^3-1\right)=3^{12}\cdot26\)
Vì: \(26⋮13\Rightarrow3^{12}\cdot26⋮13\Rightarrow3^{15}-9^6⋮13\)
--> Đpcm
\(3^{15}-9^6\)
\(=3^{15}-\left(3^2\right)^6\)
\(=3^{15}-3^{12}\)
\(=3^{12}.3^3-3^{12}.1\)
\(=3^{12}.27-3^{12}.1\)
\(=3^{12}\left(27-1\right)\)
\(=3^{12}.26\)
\(=3^{12}.2.13⋮13\Rightarrowđpcm\)
Ta có: \(3^3\equiv1\left(mod13\right)\)
\(\Rightarrow3^{15}\equiv1\left(mod13\right)\)
\(9^3\equiv1\left(mod13\right)\)
\(\Rightarrow9^6\equiv1\left(mod13\right)\)
\(\Rightarrow3^{15}-9^6\equiv0\left(mod13\right)\)
\(\Rightarrow3^{15}-9^6⋮13\)
\(\Rightarrowđpcm\)
